A measuring system for contactless or tactile measurement on tools in a machine tool is described below. These types of measuring systems are to be used in machine cutting or material-removing machining (for example, milling, turning, grinding, planning, drilling, countersinking, reaming, eroding, and the like), also in combined lathe/milling machines or milling/turning machines with stationary or rotating tools. Features and properties of the measuring system are defined in the claims; however, the description and figures also disclose characteristics of the measuring system and its various aspects.